Factorial Calculator
Calculate the factorial of any whole number from 0 to 12 instantly.
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The factorial calculator computes n! — the product of all positive integers from 1 up to n — for any whole number from 0 to 12. Select a number from the dropdown and get the result instantly. Factorials are fundamental to probability and combinatorics. Any time you are counting the number of ways to arrange or choose items — seating arrangements, card hand combinations, lottery odds, or password permutations — factorials are the underlying operation. They also appear in binomial expansions, Taylor series in calculus, and the mathematical definition of combinations and permutations. Results grow extremely fast: 1! = 1, but 12! = 479,001,600. This rapid growth (called factorial growth) is why even a modest deck of 52 cards has more possible shuffles (52! ≈ 8 × 10⁶⁷) than atoms in the observable universe — a fact that makes every shuffled deck arrangement almost certainly unique in history.
How to Use the Factorial Calculator
The Factorial Calculator is designed to give you an accurate answer in seconds. Follow these steps:
- Step 1: Choose your number (n) from the Number (n) dropdown. Select the option that most accurately reflects your current situation — this value feeds directly into the calculation.
- Step 2: Click Calculate to see your results instantly. The output updates as soon as you submit.
No account or sign-up required. All calculations run locally in your browser — nothing is stored or transmitted to any server.
Example Calculation
Here is what the Factorial Calculator produces with its default values. Change any input above to recalculate instantly for your own figures.
Inputs
- Number (n)5
Results
- Factorial (n!)5
How It Works
n! = n × (n−1) × (n−2) × ... × 2 × 1
Formula: n! = n × (n−1) × (n−2) × ... × 2 × 1 Factorial multiplies every integer from n down to 1 together: 0! = 1 (by definition) 1! = 1 2! = 2 × 1 = 2 3! = 3 × 2 × 1 = 6 4! = 4 × 3 × 2 × 1 = 24 5! = 5 × 4 × 3 × 2 × 1 = 120 10! = 10 × 9 × ... × 1 = 3,628,800 12! = 479,001,600 Why 0! = 1: by convention and because it makes the formulas for combinations and permutations consistent. The number of ways to arrange zero items in a sequence is exactly one way — do nothing. Factorials in practice: Permutations (ordered arrangements of r items from n): P(n,r) = n! ÷ (n−r)! Combinations (unordered selections of r from n): C(n,r) = n! ÷ (r! × (n−r)!) Example: ways to choose 3 cards from 5 = 5! ÷ (3! × 2!) = 120 ÷ (6 × 2) = 10
Frequently Asked Questions
What is 5 factorial (5!)?
5! = 5 × 4 × 3 × 2 × 1 = 120. This means there are 120 different ways to arrange 5 items in a sequence. For example, 5 people in a line can be ordered 120 distinct ways. Factorials count permutations — all possible orderings of a set.
Why does 0! equal 1?
By mathematical convention, 0! = 1 because it makes combination and permutation formulas consistent. There is exactly one way to arrange zero objects: do nothing. It also follows from the recursive definition of factorial: n! = n × (n−1)!, which gives 1! = 1 × 0!, so 0! must equal 1 for the formula to hold.
How are factorials used in probability?
Factorials calculate the number of possible arrangements or selections. Combinations (choosing r items from n without caring about order) use C(n,r) = n! ÷ (r! × (n−r)!). Permutations (ordered selections) use P(n,r) = n! ÷ (n−r)!. Poker hand odds, lottery probabilities, and DNA sequence counts all rely on these formulas.
Why do factorials grow so fast?
Each additional number multiplies the entire previous result. Going from 5! (120) to 6! requires multiplying by 6, giving 720. From 10! (3,628,800) to 11! requires multiplying by 11, giving 39,916,800. This multiplicative compounding means factorials grow faster than exponential functions — 20! has 19 digits, and 100! has 158 digits.
Is the factorial calculator free?
Yes — free with no sign-up needed. Select any number from 0 to 12 for an instant result. All calculations run in your browser and no data is stored.